Lorentz contraction

cfrogue

OK, sorry, when you have some time, I am not seeing your explanation.

1) One solution suggests there exists length contraction for the string.
2) One solution suggests the ships get further apart.
3) The rest frame concludes the distance remains constant between the ships and the v and any time t is the same.

Actually, if you look from the rest frame, a reaction may be that as v increases, length contraction for the string should increase.

Yet, the SR acceleration equations do not predict this and predict a constant distance between the ships.

How is this worked out?

JesseM

That would be wrong, you can only use the length contraction equation for an object with a constant length in its rest frame, but the string's length in its rest frame is changing because its ends are attached to the ships.
Well, only in the launch frame, not in other frames.

cfrogue

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

So, how would the launch frame conclude the string contracts when the launch frame concludes the distance between the ships does not change?


Understood

JesseM

In the launch frame? No, of course the string does not contract in this frame (at least not until it breaks), how could it when it's attached to the ships? Why would you think it should? Did you read what I just said about the length contraction not applying when the rest-frame length of an object is not constant? There is no question that the rest-frame length of the string in this scenario is not constant.

cfrogue

OK, then how does the string break from only the solution of the launch frame?

Al68

In a co-moving reference frame, the ships are getting farther apart with time. In the launch frame the distance between the ships stays the same. That's length contraction, by a factor that increases with time.

In a co-moving reference frame, the ships get farther apart while the length of the string stays the same. The string breaks.

In the launch frame, the ever increasing length contraction factor results in a constant distance between the ships, while the string gets "shorter". The string breaks.

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite being stretched in its own frame.

Length contraction doesn't mean contracting with time, it means contracted relative to the proper length.

JesseM

You asked this question before, I gave my answer in post #42 when I said:

JesseM

This explanation seems confused to me...why do you say it's the "length contraction factor" that results in a constant distance? And why do you say the string gets shorter? Both the distance between ships and the length of the string are constant in the launch frame, because both ships have identical velocity as a function of time in this frame.

cfrogue

Can you explain how this is consistent with the SR acceleration equations?

Al68

Because while the distance is constant in the launch frame, it is contracted by the ever increasing lorentz factor. I didn't intend to imply a causal relationship by the words "results in".That part of my post was assuming the string wouldn't stretch, and therefore break. I addressed a stretchy string in the next part:

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite its proper length increasing with time.

JesseM

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

JesseM

Presumably you mean it's "contracted" relative to the distance in some other frame, like the instantaneous rest frame of one of the ships? But this is still a bit murky, because in every other frame the distance between the ships is changing, and you can't really compare the distance between ships at a given moment in one of these frames with the distance between them in the launch frame "at the same moment" without running into simultaneity issues. I suppose we can say that if we look at the length L' in the launch frame at any given time t, and then imagined the ships shutting off their engines simultaneously at time t in the launch frame and coasting inertially thereafter, and then we looked at the length L in the inertial frame where the ships were at rest once they had both shut off their engines, then it would make sense to say that L' is related to L by the length contraction equation L' = L/gamma.

cfrogue

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

JesseM

I didn't prove it from the perspective of the launch frame, I just outlined how I think you would go about proving it in terms of changing electromagnetic forces between atoms, and then said "The details of such a calculation are beyond me though."

However, if we're just interested in the question of whether it snaps or not, we don't actually have to prove it from the perspective of multiple frames, proving it snaps using the perspective of any one frame is good enough, since in relativity all frames always agree on localized physical events.

Al68

Yes, and gamma would depend on t, which is what I meant by the lorentz factor increasing with time.

My only point was that the constant distance between the ships in the launch frame is "contracted" distance, and that "constant distance" and "increasingly lorentz contracted distance" aren't contradictory since the proper distance between the ships is increasing with time.

Al68

Are you suggesting that the string's proper length can stretch indefinitely without snapping?

cfrogue

Oh, this is where I am confused.

Where is the string snap logic in the context of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?